KK-theoretic duality for proper twisted actions

Abstract

Abstract. Let A be a smooth continuous trace algebra, with a Riemannian manifold spectrum X, equipped with a smooth action by a discrete group G such that G acts on X properly and isometrically. Then A −1 ⋊G is KK-theoretically Poincaré dual to`A⊗ C 0 (X) Cτ (X)´⋊G, where A −1 is the inverse of A in the Brauer group of Morita equivalence classes of continuous trace algebras equipped with a group action. We deduce this from a strengthening of Kasparov's duality theorem. As applications we obtain a version of the above Poincaré duality with X replaced by a compact G-manifold M and Poincaré dualities for twisted group algebras if the group satisfies some additional properties related to the Dirac dual-Dirac method for the Baum-Connes conjecture.

Twisted $K$-theory and Poincaré duality

“…Thom isomorphism, which is closely related to Poincaré duality, was established by Karoubi in [11] in the setting of non-twisted K-theory, and by Carey and Wang [5] and Karoubi [12] in twisted K-theory. Let us also mention that (a slight variation) of our main result was obtained independently and simultaneously by Echterhoff, Emerson and Kim [7].…”

Geometric cycles, index theory and twisted K-homology

“…In [22] (see also [42]), a natural isomorphism, called the Poincaré duality in analytical twisted K-theory, KK.C; C c .X; P˛.K// Ő C c .X/ C c .X; Cliff.TX/// Š KK.C c .X; P ˛. K//; C/ is constructed using the Kasparov product with the weak dual-Dirac element associated to P˛.K/; see Definition 1.11 and Theorem 1.13 in [22] for details.…”

“…K//; C/ is constructed using the Kasparov product with the weak dual-Dirac element associated to P˛.K/; see Definition 1.11 and Theorem 1.13 in [22] for details.…”

kk-Theory for Banach algebras II: Equivariance and Green–Julg type theorems

“…for all Banach algebras A and B, where X is a G-compact proper G-space and A is a certain proper G-Banach algebra; the result is, on an abstract level, a Banach algebraic analogue of the C * -algebraic Poincaré duality, compare [EEK10,EEK08,Eme03,Con94]. But there is a technical problem that one has to overcome before one can actually apply this abstract result to actions on manifolds, see Remark 6.3.…”